Displacement with a stopping object
Use 2 displacement modifiers with the same settings.


1st will displace the object until it touches the collision object
2nd will continue with displacing, but only around the object. To do that, paint and assign a vertex group to the modifier to influence only the right vertices.


Simulating this inside Blender is not possible, because Blender does not simulate pressure inside volumes. With SoftBody sim there is a possibility to alter the edge springs to make the mesh sort of "blow up" (Length% under Soft Body Edges tab), but the result is not nice and does not look like inflating.

Houdini can simulate pressure in volumes very nicely, the apprentice version is free and can import/export .obj(s) if you wanna try it.